You are imagining a situation where the treatment would be counter productive.
I know iodine is needed for thyroxine production, but according to "wolf-chaikoff" effect, iodine in high doses actually deplete the production of thyroxine
... that would be if the patient has a too-high iodine level ... the feedback process remember?
That is why increasing iodine in the diet is only useful if the patient has an iodine
deficiency.
Not all the iodine in the blood supply goes into T4 production.
If iodine levels should drop, then the percentage that goes to T4 production needs to increase to maintain production to required levels. To do this, you have to increase iodine trapping, and the amount of blood that goes through the thyroid ... which is what THS is for. THS levels normally go up more than needed. Once production is back up, then THS levels fall to toughly the new level needed to maintain something like the proper T4 supply.
But if there is a chronic shortage of iodine it does not matter how much THS is supplied, there is not enough iodine in the blood supply to make the required T4. Imagine an extreme case - no iodine available at all! The thyroid gets swollen - to no effect - and the patient also suffers the effects of not having enough T4 etc.
Clearly the solution is to add more iodine - usually via the diet.
Iodine levels increase, T4 gets made at a huge rate (because the body is geared to a bad shortage), when T4 levels reach requirement THS levels drop off, reducing T4 production to something more sustainable with the new level of iodine supply... the thyroid swelling goes down.
As the iodine levels get higher than required, then iodine trapping is reduced so a smaller proportion of iodine gets supplied - the aim being to maintain T4 production by choking off the supply. However, it's not linear ... very high iodine levels can trigger an overreaction (so to speak) and T4 production drops too far for comfort.
The normal healthy process will operate in a quasi-linear region where the feedback works sort-of smoothly.
The process can get unhealthy for any number of reasons - so if you are being skeptical of claims that all thyroid problems can be addressed by iodine supplements then you are right to be so. It is possible to have too much iodine.
Fortunately - "low iodine levels" is a likely cold-read, like "I feel there's someone with a C who has passed on..." and, for a healthy person, the supplements/remedies won't do much of anything ... cue the placebo effect and you have a pretty standard pseudoscience treatment.